BERNSTEIN CLASSES

One of the classical Bernstein inequalities compares the maxima of a p olynomial of a given degree on the interval [-1,1] and on the ellipse in the complex plane with the focuses -1,1 and the semiaxes R. We prov e a similar inequality for a branch of an algebraic function of a give n degree on the maximal disk of its regularity, with the explicitly gi ven constant, depending on the degree only. In particular, this improv es a recent inequality of Fefferman and Narasimhan and answers one of their questions. We present in detail various properties of the classe s of functions, satisfying Bernstein type inequalities and various app roaches to establishing such inequalities.